Introduction

The analysis of the independence of random variables, particularly (X Y) and (X - Y), is a critical concept in probability theory and statistics. Understanding these relationships can provide deep insights into the behavior of these variables and their joint distributions. This article explores how the independence of (X Y) and (X - Y) can be determined, utilizing characteristic functions as a powerful tool. We will delve into the underlying mathematical transformations and distributions involved.

Transformations and Variables

Consider two random variables, (X) and (Y). For the purpose of this analysis, let's assume (X) and (Y) are independent and identically distributed (i.i.d.) continuous random variables. We are interested in the variables (U X Y) and (V X - Y). To analyze the independence of these new variables, we will use random variable transformations and Jacobians.

Transformations

We define the transformations as follows: (U X Y) (V X - Y) Expressing (X) and (Y) in terms of (U) and (V), we get: (X frac{U V}{2}) (Y frac{U - V}{2})

Jacobian Determinant

The Jacobian of the transformation from (X, Y) to (U, V) is a crucial component in determining the joint distribution of (U) and (V):

[text{Jacobian} begin{pmatrix} frac{partial U}{partial X} frac{partial U}{partial Y} frac{partial V}{partial X} frac{partial V}{partial Y} end{pmatrix} begin{pmatrix} 1 1 1 -1 end{pmatrix}]

The determinant of this Jacobian is:

[text{det} 1 cdot (-1) - 1 cdot 1 -2]

The absolute value of the Jacobian is thus 2.

Joint Distribution of (U) and (V)

The joint density function of (U) and (V) can be expressed using the joint density of (X) and (Y). Given that (X) and (Y) are i.i.d. and have a joint distribution (f_{XY}(x, y)), the joint density of (U) and (V) is:

[f_{UV}(u, v) f_{XY}left(frac{u v}{2}, frac{u - v}{2}right) cdot |text{Jacobian|} f_Xleft(frac{u v}{2}right) f_Xleft(frac{u - v}{2}right) cdot 2]

For simplicity, assume (X) and (Y) follow a normal distribution. The normality assumption simplifies the joint density function and allows us to further analyze the independence of (U) and (V).

Independence Condition

To check the independence of (U) and (V), we need to verify if the joint density can be factored into the product of marginal densities. For (U) and (V) to be independent, the joint density must satisfy:

[f_{UV}(u, v) f_U(u) f_V(v)]

This condition holds if and only if the joint distribution can be expressed as a product of two functions, one depending only on (u) and the other only on (v). In the context of characteristic functions, this independence is equivalent to the characteristic functions of (U) and (V) being the product of the characteristic functions of (X) and (Y).

Characteristic Functions

The characteristic function of a random variable (Z) is defined as (phi_Z(t) E[e^{itZ}]). For the random variables (U) and (V): (phi_U(t) phi_X(t) phi_Y(t)) (phi_V(t) phi_X(t) phi_Y(-t) phi_X(t) phi_X(-t)) Given the normality of (X) and (Y), the characteristic functions simplify to (phi_X(t) exp(-t^2/2sigma^2)), leading to: (phi_U(t) (exp(-t^2/2sigma^2))^2 exp(-t^2/sigma^2)) (phi_V(t) (exp(-t^2/2sigma^2))^2 exp(-t^2/sigma^2)) Since the characteristic functions of (U) and (V) are the same as that of a single normal random variable (X), and they factor into the product of the characteristic functions of (X) and (Y), it confirms the independence of (U) and (V) when (X) and (Y) are independent and identically distributed normal random variables.

Summary

In summary, (X Y) and (X - Y) are independent if and only if (X) and (Y) are independent and identically distributed normal random variables. Understanding this relationship can be crucial in various applications of probability theory and statistics.